Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvat | Structured version Visualization version GIF version | ||
| Description: If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 32113 analog.) (Contributed by NM, 11-Jan-2015.) |
| Ref | Expression |
|---|---|
| lcvat.s | β’ π = (LSubSpβπ) |
| lcvat.p | β’ β = (LSSumβπ) |
| lcvat.a | β’ π΄ = (LSAtomsβπ) |
| icvat.c | β’ πΆ = ( βL βπ) |
| lcvat.w | β’ (π β π β LMod) |
| lcvat.t | β’ (π β π β π) |
| lcvat.u | β’ (π β π β π) |
| lcvat.l | β’ (π β ππΆπ) |
| Ref | Expression |
|---|---|
| lcvat | β’ (π β βπ β π΄ (π β π) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lcvat.s | . . 3 β’ π = (LSubSpβπ) | |
| 2 | lcvat.p | . . 3 β’ β = (LSSumβπ) | |
| 3 | lcvat.a | . . 3 β’ π΄ = (LSAtomsβπ) | |
| 4 | lcvat.w | . . 3 β’ (π β π β LMod) | |
| 5 | lcvat.t | . . 3 β’ (π β π β π) | |
| 6 | lcvat.u | . . 3 β’ (π β π β π) | |
| 7 | icvat.c | . . . 4 β’ πΆ = ( βL βπ) | |
| 8 | lcvat.l | . . . 4 β’ (π β ππΆπ) | |
| 9 | 1, 7, 4, 5, 6, 8 | lcvpss 38397 | . . 3 β’ (π β π β π) |
| 10 | 1, 2, 3, 4, 5, 6, 9 | lrelat 38387 | . 2 β’ (π β βπ β π΄ (π β (π β π) β§ (π β π) β π)) |
| 11 | 4 | 3ad2ant1 1130 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β LMod) |
| 12 | 5 | 3ad2ant1 1130 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β π) |
| 13 | 6 | 3ad2ant1 1130 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β π) |
| 14 | simp2 1134 | . . . . . . 7 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β π΄) | |
| 15 | 1, 3, 11, 14 | lsatlssel 38370 | . . . . . 6 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β π) |
| 16 | 1, 2 | lsmcl 20927 | . . . . . 6 β’ ((π β LMod β§ π β π β§ π β π) β (π β π) β π) |
| 17 | 11, 12, 15, 16 | syl3anc 1368 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β (π β π) β π) |
| 18 | 8 | 3ad2ant1 1130 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β ππΆπ) |
| 19 | simp3l 1198 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β π β (π β π)) | |
| 20 | simp3r 1199 | . . . . 5 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β (π β π) β π) | |
| 21 | 1, 7, 11, 12, 13, 17, 18, 19, 20 | lcvnbtwn2 38400 | . . . 4 β’ ((π β§ π β π΄ β§ (π β (π β π) β§ (π β π) β π)) β (π β π) = π) |
| 22 | 21 | 3exp 1116 | . . 3 β’ (π β (π β π΄ β ((π β (π β π) β§ (π β π) β π) β (π β π) = π))) |
| 23 | 22 | reximdvai 3157 | . 2 β’ (π β (βπ β π΄ (π β (π β π) β§ (π β π) β π) β βπ β π΄ (π β π) = π)) |
| 24 | 10, 23 | mpd 15 | 1 β’ (π β βπ β π΄ (π β π) = π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwrex 3062 β wss 3941 β wpss 3942 class class class wbr 5139 βcfv 6534 (class class class)co 7402 LSSumclsm 19550 LModclmod 20702 LSubSpclss 20774 LSAtomsclsa 38347 βL clcv 38391 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-cntz 19229 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-ur 20083 df-ring 20136 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lsatoms 38349 df-lcv 38392 |
| This theorem is referenced by: islshpcv 38426 |
| Copyright terms: Public domain | W3C validator |